transformations transformations of functions and graphs we will be looking at simple functions and...
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TransformationsTransformationsTransformations
Transformations
Transformations of Functions and Graphs
We will be looking at simple functions and seeing how various modifications to the functions transform them.
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Above is the graph of 2xxf
x
y
What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).
x
y
11 2 xxf
What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them).
x
y
33 2 xxf
As you can see, a number added or
subtracted from a function will
cause a vertical shift or
translation in the function.
2xxf
y = f(x) + k
y = f(x) ̶ k
Transformation
k units
Down k units
UP
VERTICAL TRANSLATIONS
)(xg
)(xh
xf
3xf
3xf
what is the transformation?
y = f(x) +10
Parent function
y = f(x) Up 10 units
Down 9 units
Up 5 units
Down 7 unitsy = f(x)
Above is the graph of 2xxf
x
y
What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function).
As you can see, a number added or
subtracted from the x will cause
a horizontal shift or
translation in the function but opposite way of the sign of the
number.
HORIZONTAL TRANSLATIONS
x
y
x
y 2xxf
211 xxf
222 xxf
What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).
y = f(x + h)
y = f(x ̶ h)
Transformation
h units
Right h units
left
For h>0, and
HORIZONTAL TRANSLATIONS
x
y
xf
)3( xfshift right 3
1xf
)(xh
)(xgshift left 1
what is the transformation?
y = f(x+10)
Parent function
y = f(x) Left 10 units
Right 9 units
Left 5 units
Right 7 unitsy = f(x)
x
y
x
y
x
y
We could have a function that is transformed or translated both vertically AND horizontally.
Above is the graph of xxf What would the graph of look like? 3)2( xxf
up
3
left 2
what is the transformation?
y = f(x+1)-6
Parent function
y = f(x) Left 1 and down 6
Right 3 and up 2
Left 5 and up 7
Right 8 and down 1 y = f(x-8)-1 y = f(x)
and
If we multiply a function by a non-zero real number it has the effect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number.
Let's try some functions multiplied by non-zero real numbers to see this.
DILATION:
Above is the graph of xxf
So the graph a f(x), where a
is any real number
GREATER THAN 1, is the graph of f(x)
x
y
x
y
x
y
xxf
xf2 xxf 44
What would 2f(x) look like?
What would 4f(x) look like?
Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value.
Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.
The bigger a is. The narrower
the graph is.
vertically
stretched by a factor of a.
x2
Above is the graph of xxf
So the graph a f(x), where a
is 0 < a < 1, is the graph of f(x)
x
y
x
y
Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value.
x
y
Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value.
xxf4
1
4
1
What if the value of a was positive but less than 1?
xxf
xxf2
1
2
1
What would 1/4 f(x) look like?
What would 1/2 f(x) look like?
The smaller a is.
The wider the graph is.
verticallyCompressed
by a factor of a.
y = a f(x) Transformation
Stretched Vertically
By factor of a a>1
0<a<1 Compressed
Vertically
By factor of a
VERTICAL TRANSLATIONS
)(2 xf
xf
xf4/1
xf2/1
3. Horizontal translation
Procedure: Multiple Transformations (From left to right)
2. Stretching or shrinking
1. Reflecting
4. Vertical translation
what is the transformation?
y = 5f(x+10)-6
vertically stretched by factor of 5, Left 10, down 6
vertically compressed by factor of ¼ , Right 7, up 2
what is the transformation?
vertically compressed by factor of 1/5, Left 6, down 7
vertically stretched by factor of ¼ , Right 9, up 2
Above is the graph of xxf
So the graph - f(x) is a reflection about the
x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the
function over the x-axis)
x
y
What if the value of a was negative?
What would - f(x) look like?
x
y
xf
xxf
Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value. x
x
y
Above is the graph of 3xxf
There is one last transformation we want to look at.
Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value.
x
y
3xxf xf
What would f(-x) look like? (This means we are going to take the negative of x before putting in the function)
So the graph f(-x) is a
reflection about the
y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the
function over the y-axis)
3x
Summary of Transformations So Far
khxfa h >0
vertically stretched by a factor of a
Up k units
vertically compressed by a factor of a
reflected across y-axis
**Do reflections and dilations BEFORE vertical and horizontal translations**
-f (x)
If a > 1,
If 0 < a < 1,
reflected across x-axis
f(-x)
(opposite sign of number with the x)
Left h units
h <0 Right h units
k>0
k<0 Down k units
Graph using transformations 12
1
xxf
We know what the graph would look like if it wasfrom our library of functions.
x
xf1
x
y
moves up 1
moves right 2
reflects about the
x -axis
x
y
x
y
x
y
x
y